Monstrous moonshine [1]} assigns to every conjugacy class of an element \(g\)  of the monster sporadic simple group \(\mathbf {M}\)  a specific genus zero subgroup \(\Gamma _g\)  of \(PSL_2(\mathbb {R})\)  which is commensurable with the modular group \(\Gamma \)  (that is, \(\Gamma _g \cap \Gamma \)  has finite index in \(\Gamma _g\)  and \(\Gamma \) ). There are exactly 171 such moonshine groups, usually given in Conway-Norton notation
\(\Gamma _g = n|h+e,f,\hdots \) 
